We prove some Schauder-type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.
"Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form." Adv. Differential Equations 11 (11) 1261 - 1320, 2006.